The Mixed Hodge structure on the relative cohomology is defined by means of the mixed cone (see [1]}). The important part is that the long exact sequence of the pair ends up being a long exact sequence of mixed Hodge structures and so the Hodge and the weight filtration may be defined via this sequence. Also, the Gauss-Manin connection satisfies the Griffths transversality property for the Hodge filtration, that is, \(\nabla (F^p) \subset F^{p-1}\otimes \Omega ^1_S\) . This is easily seen using that fact that the Gauss-Manin connection commutes with the long exact sequence and that the long exact sequence is a sequence of mixed Hodge structure. We have a nice description of the mixed Hodge structure for our case, given by the proposition below.
